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CHEM 230 (5)
Lecture

Class Notes 3 Summer 2012.pdf

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Department
Chemistry
Course
CHEM 230
Professor
Jennifer Warriner
Semester
Summer

Description
4. Molecular Orbital Theory– • Odd number of bonding e : H - + → Stable s 2 • O , VB →All e are paired ⇒ should be diamagnetic 2 s • Experiment: Paramagnetic, 2 unpaired e ! s- • Difficulties for more complex molecules • O ,3coordination compounds, benzene • Cannot see bond energy levels via spectroscopy ⇒ Need for a more sophisticated approach⇒ MO Theory (a)Formation of Molecular Orbitals • Linear Combination ofAtomic Orbitals(LCAO) (i) Single electron molecule: H 2+ Schrodinger Eqn. (in atomic units; 1au ≈27.2eV) 2 [-1/2∇ –1/r – a/r + b/R]Ψ= EΨ Molecular wave function (WF)Ψ⇒ Molecular orbitals Atomic orbitals ↔ wave function Ψ and Ψ a b Wave properties of e ⇒suserposition of atomic waves • LCAO ⇒Ψ= c Ψ + c a a b b • ca& c : beighting functions • Two wave functions ⇒Two molecular orbitals with distinct E • Ψ = Ψ + Ψ ⇒ E = f (R) →Min. at equilibrium distance R 1 a b 1 1 o • Ψ = Ψ – Ψ ⇒ E = f (R) →Monotonically ↑when R ↓ 2 a b 2 2 34 • R o 1.04 Å , E 1min= -269 kJ/mol Ψ → Bonding orbital , E < E ⇒ Stable 1 1 H Ψ →Antibonding orbital , E > E 2 2 H Note* Total orbital energies unchanged LCAO: Ψ= c Ψ + a Ψ a c b +…b c c • N atomic orbitals⇒ N molecular orbitals • ≈N/2 ⇒ Constructive: Bonding orbitals (E ↓) • ≈N/2 ⇒ Destructive: Antibonding orbitals (E↑) (b) Covalent Bonding Revisited (i) Function of the density of probability - • Bonding orbital Ψ ⇒ 1igher e density between two atoms - ⇒Attraction of e by nuclei ⇒ H more stable than (H + H) or (H + H ) by bonding 2 interaction - • Antibonding orbital Ψ 2 ⇒lower inter-nuclear e density ⇒ (ii) Covalent bonding • Overlap of atomic orbitals ⇒ Bonding (molecular) orbitals(BOs) - • es→ BOs ⇒ (c) Molecular Orbitals •Multielectron molecule: (one)i –e → Ψ - HΨ = EΨ i i i i i •Bonding e ⇒ over whole molecule ⇒ delocalized approach s 35 Total WF: Ψ= Ψ Ψ ….Ψ Total energy E = E + E +….E 1 2 n 1 2 n (d) Conditions for Molecular Orbital Formation (i) Close energy levels for atomicorbitals • Outermost orbitals with small∆E ⇒ • If ∆E (Ψ bΨ ) as large • E ≈ Ψ and E ≈ Ψ ⇒ no bonding 1 a 2 b (ii) Maximum overlap of atomic orbitals • Overlap integral β = • Directionality of covalent bonding (iii) Matching of orbital symmetry w.r.t. to molecular axis Positive overlap • Same signs for superimposed regions of 2 orbitals • Signs of wavefunctions ⇒[(+ve) + (+ve)] or [(-ve) + (-ve)] ⇒ e-density ↑ ⇒ E ↓ Negative overlap ⇒ non-matching of orbital axial symmetry • Opposite signs for the two superimposed regions ⇒ [(+ve) + (-ve)] ⇒e-density ↓ ⇒ E ↑ ⇒Antibonding interaction (Orbitals*) Zero overlap Equal regions of opposite overlap ⇒[Bonding + Antibonding] →∆E = 0⇒ 36 (e) Electronic Configuration of Molecular Orbitals • Three rules 1. The aufbau principle ⇒ 2. Pauli exclusion principle 3. Hund’s rule of max. multiplicity • Information on Antibonding Orbitals (ABOs) • [Bonding +Antibonding] ⇒ • Order of E-levels: • ABOs ⇒ same properties as BOs: • Important role in chemical bonding and molecular excitation + + • Examples H , H2, He 2 He 2 2 (f) Bond Order • Bond order = (# of bonding e - # antibonding e )/2 - s s • Larger bond order ⇒ greater bond strength + + • H 2 0.5 < H = 1 2 He = 0.5 <2He = 0 2 (g) Classification of Molecular Orbitals • Atomic orbitals: s, p, d ⇒ molecular orbitals: σ, π, δ (i) σ-orbitals and σ-bond • No change in sign & shape when rotated around bond axis • Formed by both +ve & -ve overlap ofAO of same axial property (s + s’): +ve overlap ⇒ σ ; -ve overlap (s-s’) ⇒ σ * s s (p z p ’z: +ve overlap ⇒ σ ; -vepoverlap (p –p ’) ⇒ z * z p (s + p z): +ve overlap ⇒ σ ; -ve overlap (s –p ’) ⇒ σz 37 • Head-to-head overlap • e in the σ-orbitals ⇒ s (ii) π-orbitals and π-bond • WF separated by two regions of opposite signs • Anodal plane containing the bond axis ⇒ • π-MO ⇔ p-AO, formed by two p-orbitals (p + p ’): +ve overlap ⇒ π ; -ve overlap (p –p ’) ⇒ π * x x x x x x (py+ p ’y: +ve overlap ⇒ π ; -veyoverlap (p –p ’) ⇒yπ *y y ⇒ Two equivalent bonding MOs: ⇒ Two equivalent antibonding MOs: π * & π * x y nd • A 2 nodal plane ⊥ bonding axis for π * & π *x⇒ E ↑ y (iii) δ-orbitals and δ-Bond • Two nodal planes through the bond axis • Formed by overlap of suitable d-orbitals • ex. (dxy d ’)xylong z-bonding axis 4.2 MOs of HomonuclearDiatomic Molecules (a) Diatomic Molecules H and He2 2 • Combination of s-orbitals • (1s + 1s’) → σ (-1s) ; b1s - 1s’) → σ * (+E1s b • Electronic configurations: • H : (σ ) ; bond order = (2-0)/2 = 1 ⇒ 2 1s • He 2 38 (b) General Description of Diatomic molecules(1 short period) st • Inner shell e (ss ): closer to nuclei→ non-bonding • Only valence orbitals → overlap ⇒ (i) Combination of s-orbitals • (2s + 2s’) → σ ; (2s - 2s’) → σ * 2s (ii) Combination of p-orbitals: 2x (2p , 2p ,x2p ) y z • (2p z 2p ’)z→ σ ; (2p2p 2p ’) z σ * z 2p • (2p + 2p ’) → π ; (2p + 2p ’) → π x x 2px y y 2py ⇒ Doubly degenerate 2x π at E(π ) 2p 2p • (2p - 2p ’) → π * ; (2p – 2p ’) → π * x x 2px y y 2py ⇒ Doubly degenerate 2x π * at E(π *) 2p 2p • 2x (p x p ,yp ) z 6 MOs with 4 energy levels (c) MO Energy levels (without s-p mixing) • E(s) < E(p) ⇒ E(σ & σ *2s< E (2s) 2p • σ-orbitals → larger overlap as compared toπ-orbitals ⇒ Greater energy split⇒ ∆E(σ *- σ ) > ∆2pπ *- 2p) 2p 2p E(σ )< E(σ *)< E(σ )< E(π )= E(π )< E(π *)= E(π *)< E(σ *) 2s 2s 2p 2px 2py 2px 2py 2p 39 (i) The F m2lecule: 2x(1s 2s 2p ) ⇒ 14 valence e s- [He ]2= (σ ) 1s *) 1s 2 ⇒[He ] (2 ) (2s*) (σ 2s(π ) (2p*) (σ 2p 4 2p 4 2p 0 • Bond order = • F 2olecule ⇒ a single bond F-F • Six non-bonding pairs ⇒ Three lone pairs for each F 2 2 4 - (ii) The O mo2ecule: 2x(1s 2s 2p ) ⇒ 12 valence e s 2 2 2 4 1 1 0 ⇒[He ] (2 ) (2s*) (σ 2s(π ) (2p 2p 2px*) (π *2pyσ *) 2p - • Two unpaired e ⇒ s • Bond order = (8-4)/2 = 2 ⇒ Two bonding pairs : 2 2 1 - (iii) The B m2lecule: 2x(1s 2s 2p ) ⇒ 6 valence e s 2 2 2 ⇒[He ](σ2) (σ2s) (σ 2s→ 2p - • Experiment shows paramagentic, 2 unpaired e • Solution? (d) Mixing (Hybridization) of the 2s and 2porbitals (i) When E(2s)≈ E(2p), 2s and 2p (bond axis) orbitals overlap z ⇒ [σ * and σ ] ⇒ both 2s and 2p character 2s 2p ⇒New MO sets; Hybrid MOs 40 (ii) Symmetry (Parity) of the MOs –Anew notation • Under operation of a center of inversion • An MO remains unchanged (same sign) ⇒ • An MO changes sign (opposite sign) ⇒ • In general, σ →σ ; g*→σ * ; πu→π ; π* →u * g • σ →2σ ; σ * →2σ * ; σ →3σ ; σ * →3σ * 2s g 2s u 2p g 2p u • π2px = π2py → 1π ;u[1π →udoubly degenerate] • π2px* = π 2py 1π * ;g[1π * → goubly degenerate] (iii) Valence Orbital Potential Energies Li Be B C N O F Ne E(2s) -5.5 -9.3 -14.0 -19.5 -25.5 -32.4 -46.4 -48.5 E(2p) -8.3 -10.7 -13.1 -15.9 -18.7 -21.6 ∆E(2p-2s) 5.7 8.8 12.4 16.5 27.7 26.9 • Smaller ∆E (2p-2s) for Li, Be, B, C, and N ⇒ Stronger 2s-2p interaction ⇒ Hybrid MO energy levels ⇒ • Larger ∆E (2p-2s) for O, F, and Ne ⇒ Weaker 2s-2p interaction ⇒ No 2s-2p mixing ⇒ E(3σ ) < E(1π ) ↔ [E(σ )< E(π )] 41 g u 2p 2p s-p Mixing in detail: • Take MO diagram without s-p mixing • To form MOs, mix AOs of similar energy and same symmetry • Results in 1. More stable bonding orbital 2. Less stable antibonding orbital • Two new MOs from 2AOs • In most molecules s+p orbitals do not mix due to large energy difference • Across period Z increases: eff • Greater influence on s-orbitals as compared to p-orbitals • ∆E(2p-2s) increases from B→ F • ∆E(2p-2s) small from Li→ N • Orbitals of same symmetry/similar energy can mix ⇒ ∴ 2σ /g3σ ang 2σ * / 3u * canumix • 1π u 1π g have π-symmetry so cannot mix with σ-symmetry s-orbitals – energy is unchanged • How to mix? Make new MOs…… σ (s) + σ (p ) → more stable new 2σ g g z g σg(s) - σ gp z → less stable new 3σ g 42 Changes to Orbital Interaction Diagram: • 1π and 1π * don’t change u g (iv) MO energy levels arising from s-p hybridization ⇒The most characteristic: E(3σ g) > E(1π ) u [E(σ ) > E2p )] 2p E(2σ )g< E(2σ *) u E(1π ) [xu] < E(3σ ) < E(gπ *) [x2] g E(3σ *) u E(σ )< E(σ *)< E(π ) = E(π ) < E(σ )< E(π *)= E(π *)< E(σ *) 2s 2s 2px 2py 2p 2px 2py 2p (e) The Complete Series (1 short period) (i) The Li mo2ecule: 2 valence e s- • [He ]2(2σ ) g Bond order = (2-0)/2 =1 ; Single bond (ii) The Be m2lecule: 4 valence e s- • [He ]22σ ) g2σ *) ;uBond order = 43 - (iii) The B m2lecule: 6 valence e s 2 2 1 1 • [He ]22σ ) g2σ *) (uπ ) (1πu) u - • Two unpaired e in (sπ ) ⇒ u • Bond order = (4-2)/2 = 1 - (iv) The C mo2ecule: 8 valence e s 2 2 2 2 • [He ]22σ ) g2σ *) (uπ ) (1πu) u • Bond order = (6-2)/2 = 2 2 2 2 1 1 • Excited state: (2σ ) (gσ *) (uπ ) (1π u (3σ )u g - (v) The N mo2ecule: 10 valence e s 2 2 2 2 2 • [He ]22σ ) g2σ *) (uπ ) (1πu) (3σ )u g • Bond order = (8-2)/2 = 3 • One (2σ ) bond and two (1π ) bonds → Triple bond g u • Highest bond order, shortest bond length (L = 1.10 Å) and b strongest bond: E b= 941 kJmol -1 How to describe N by MO2theory? - (vi) The Ne mo2ecule: 16 valence e s 2 2 2 4 4 2 • [He ]22
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